Questions on the mathematics behind the famed toy invented by Ernő Rubik.

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Are all sets of $n$, s.t. $R(m)^n=I$, where $R(m)$ is any sequence of $m$ moves on a Rubik's cube and $I$ is the identity operator, known?

I've written a program that finds the number of times, $n$, one must apply any operation $R_i(m)$, which consists of $m$ single moves/turns/elementary operations on a Rubik's cube, s.t. $R_i(m)^n=I$, ...
0
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1answer
15 views

Will $n$ for $A^n=\mathbb{I}$, where $A$ is any finite operation on a finite group and $\mathbb{I}$ is the identity operator, always be finite?

Will $n$ for $A^n=\mathbb{I}$, where $A$ is any finite operation on a finite group and $\mathbb{I}$ is the identity operator, always be finite? Consider for instance a finite sequence of moves (...
0
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0answers
29 views

requirements for two rubik cubes to be identical

For two Rubik cubes to be identical, what is the minimum requirements? for example are 2 faces and their position sufficient? e.g. 2 side by side face, or two opposing faces of two cubes being same ...
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0answers
14 views

About those equivalences in games, is it posible to…

I've heard about some games which can be reduced to TicTacToe, i guess in general there should be some simplier games and other complex games could be reduced to them. I've been wondering, is it ...
16
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2answers
630 views

Minimal generating set of Rubik's Cube group

The Rubik's Cube group is generated by the six moves $\{F,B,U,D,L,R\}$. However, is this the minimal generating set for the group? In other words, can I simulate the move $F$ just by making the moves $...
5
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2answers
137 views

Conjugates and commutators for twisty puzzles — so what?

This question isn't just rhetorical. I want to know what I'm missing. Twisty puzzle tutorials keep talking about how useful conjugates (operation sequences of the form ${XYX}^{-1}$) and commutators ($...
2
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0answers
102 views

Using Burnside's Lemma in GAP to handle special variations of the Rubik's Cube?

If you want to count the number of distinct positions of a standard 2x2x2 Rubik's Cube simple counting arguments will suffice: There are 8 corners, all distinct The 8 corners can be in any ...
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56 views

$2 \times2$ Rubik's cube

Let the face of the $2 \times2$ Rubik's cube be defined as the ordered quadruple of colors of it's pieces in a fixed direction (let's say counterclockwise). Does the set of $6$ quadruples define a ...
0
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2answers
45 views

No of unit cubes in a n*n*n cube

Imagine a cubic array made up of an $n\times n\times n$ arrangement of unit cubes: the cubic array is n cubes wide, n cubes high and n cubes deep. A special case is a $3\times3\times3$ Rubik’s cube, ...
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33 views

Is it possible to define a dual for the Rubiks Cube Problem?

The $3 \times 3$ Rubiks cube is already solved algorithmically. What I intended to do was to define Rubiks cube as a Linear Program. I found this link when I was searching for a formulation for it. Is ...
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35 views

Write an element of a group as product of generators

Say we have a finite group $G$ generated by $g_1,\cdots,g_n$. Are there any algorithms or techniques to write any element $g$ as a product of these generators? Ofcourse we could just try all ...
9
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1answer
203 views

Obscuring squares of Rubik's cube

This is a combinatorial question related to Rubik's cube $3\times3\times3$ (and, in the end, its generalizations $n\times n\times n$). I assume that the readers are familiar with this puzzle. Let's ...
4
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1answer
66 views

Running through all permutations of a Rubik's cube

According to Wikipedia a $3 \times3\times3$ Rubik's cube has $43252003274489856000$ permutations. I never tried solving one myself (too tedious), however I wondered, if one could miraculously "solve"...
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2answers
117 views

How to create a Rubik's cube type of puzzle that is infeasible to solve? [closed]

Is it possible to create a mechanical puzzle, something like the Rubik's cube, that is infeasible to solve in general, but still demonstrably solvable if one knows the scrambling moves. By "...
0
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0answers
23 views

All equivalent moves on a rubik's cube

Call the primitive moves on the rubik's cube "R,L,U,D,F,B" for right, left, up, down, front, and back respectively. Let us say that I have a permutation of the stickers on the cube written as a word ...
6
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1answer
159 views

Proof that the Rubik’s Cube group is 2-generated

Singmaster (1981) writes, on page 32 of his Notes on Rubik’s Magic Cube: Frank Barnes observes that the group of the cube is generated by two moves: \begin{align*} \alpha &= L^2 B R D^{-1} L^...
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3answers
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Random solving of a Rubik cube .

After playing a little with a Rubik cube I thought of the following problem : Suppose we start with a solved Rubik cube (a general one , with $n^3$ cubes) . Then we choose one of the moves , each ...
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1answer
147 views

Solving a Rubik's cube in 20 moves or less

Most "cubers" solve a Rubik's cube in about 60 moves, using special algorithms and quick fingers. But any Rubik's cube can be solved in 20 moves or less (cube20.org). Is it possible that there is a ...
7
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1answer
132 views

What are some Group representation of the rubik's cube group?

The Rubik's cube corresponds to valid sequences of moves of the Rubik's cube. What are some group representations of this group (with respect to finite dimensional vector spaces on finite fields)? ...
5
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2answers
174 views

Rubik's Revenge Cube in GAP

I'm trying to create the Rubik's Revenge (4x4x4 cube) group in GAP . Take the following net of the 4x4x4 cube with each sticker labelled with a number. The front, left, upper, right, down, and back ...
0
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1answer
30 views

Can I place the rubiks3cube pieces in the distorted position I intend to get?

Right now I am trying to get the distorted position like this: in each face only one diagonal is solved and no similar colour is on a face other than the diagonal pieces mentioned previously. For ...
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0answers
83 views

Rubik's Cube with a missing color

Suppose a color of the Rubik's cube(3x3) is missing, is it possible to find the missing color? Now a brute force method would be solve for all other colors, but that is quite naive.
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101 views

When solving a big Rubik cube (100x100x100), do you reduce the solution to like 50x50x50, and then 25x25x25, and then like 10x10x10 and then 3x3x3?

My question is about Rubiks cube. Say you're solving a 100x100x100 cube (you can see examples in youtube by computer program - https://www.youtube.com/watch?v=0cedyW6JdsQ) When solving a big Rubik ...
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19 views

Probability function for centerpieces of a 4×4 Rubik's cube

I have a question regarding blindsolving of a 4×4 Rubik's cube (a.k.a 4-blind). When a 4-blind solve begins, one is free to pick any rotation of the 4×4 cube. To minize effort needed to solve the cube,...
2
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1answer
167 views

Asymptotic value of permutations of general Rubik Cube

I found this on C|NET, and wondering if there was anything like a $1000\times1000\times1000$ cube? And how many arrangements would it have? Or if this is too much, how about an asymptotic formula for ...
2
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1answer
45 views

Homomorphism from Rubik's Cube group to Symmetric Group.

$R^2$ is the rotation of the right side by 180, $F^2$ is the rotation of the front by 180. Other than this I am completely lost and this blowing my mind. I wish I had more to build off of, but I am ...
2
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1answer
83 views

The mathematics underlying Rubik's games

I am interested in knowing (a little more) about the mathematics underlying some of Erno Rubik's games. I guess we all know his famous cube. I heard at some point that a solution was possible because ...
3
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2answers
78 views

Distance between two positions of the rubik cube .

It's now known that to reach the solved position of the rubik cube from any other position you need at most $20$ moves (a $180^{\circ}$ of a face is counted as one move , not two ) . See for example ...
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1answer
215 views

Solving a Rubik's Cube Using Calculus or differentiation? [closed]

I have this question to solve a Rubik's cube using calculus or differentiation and then figuring out whats the shortest time it could be solved in...please help me???
1
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1answer
133 views

What do I need to learn to solve the math behind a Rubiks cube?

I stopped studying math at precalc. What should I study to be able to understand how to solve a Rubics cube using mathematics? I don't think this is an opinion question, and I really want to learn ...
0
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2answers
726 views

How many ways are there to solve a Rubiks cube? [closed]

Friends have been arguing that a rubiks cube could never be mastered cause there are too many different ways to solve it cause there are 43,252,003,274,489,856,000 positions the Cube could have, so ...
0
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1answer
236 views

How many combinations in 10x10x10 Rubik's cube?

I was wondering how many possible combinations there is in the cubes greater than 3x3x3 (4x4x4, 5x5x5, ..., 10x10x10)? We know that in 3x3x3 there are about 4,3 * 10^19 combinations, what about bigger ...
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2answers
87 views

Rubik's Cube's Group

Is there an article somewhere with an exhaustive study of the Rubik's Cube Group $G$? Such as computing some subgroups of it or exhibiting some elements of its center $Z(G)$? I tried googling it and ...
2
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1answer
316 views

Rubik's Cube Thistlethwaite Four Phase Algorithm

I am trying to understand The Four-Phase-Algorithm ( Thistlethwaite's 52-move algorithm ) for Rubik's Cube but no easy understandable explaination is given. It would be helpful if given with a small ...
0
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1answer
40 views

how to evaluate combinations of different configurarations of a rubik's cube

how to calculate the total combination number of an rubik's cube ? and what is the number of combination if we fix one face of it with all matched colours , specifically saying blue one ,?? no. of ...
15
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1answer
162 views

$45^\circ$ Rubik's Cube: proving $\arccos ( \frac{\sqrt{2}}{2} - \frac{1}{4} )$ is an irrational angle?

I've been working on a problem related to the 3x3x3 Rubik's Cube where you allow faces to be turned by $45^\circ$ instead of just the usual $90^\circ$. We know for the standard 3x3x3 the cube is ...
1
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1answer
62 views

Can solution to Rubik's cube be seen from the point of view of Markov Decision Process?

Solving Rubik's cube can be thought of as a Planning problem which has : a state space $S$ a set $G \subseteq S$ of goal states (in this case singleton) actions $A(s) \subseteq A$ applicable in ...
2
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1answer
37 views

Imitating smaller Rubik's cubes with bigger ones.

Let's assume we have a 4x4x4 Rubik's cube. With this cube we can imitate a 2x2x2 by considering all 2x2x2 corner cubes as 1 block. Similarly, we can solve it like a 3x3x3 by first solving the 2x2 ...
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2answers
420 views

Number of moves necessary to solve Rubik's cube by pure chance

Suppose, random moves are made to solve Rubik's cube. A move consists of a $90$-degree-rotation of some side. The starting position is also random. What is $E(X)$, where $X$ is the number of moves ...
4
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1answer
152 views

How to learn about the Rubik's Cube?

So I just started learning Abstract Algebra and I've recently appreciated an introduction to group theory. Among the interesting avenues of group theory is found in tackling a Rubik's Cube. I've ...
0
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1answer
59 views

Help with proving property of Rubik's cube.

I am looking at these notes, and am trying to prove the following on page 13: Let $C_1$ and $C_2$ be two different unoriented corner cubies, and let $C'_{1}$ and $C'_{2}$ be two different ...
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1answer
203 views

Solve All Sequence (Rubik's Cube)

Can you prove/disprove that there is a solve-all sequence of moves to complete the Rubik's Cube from any solvable-position? If so, can you explain how long it is? If not, explain why not. Just to be ...
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2answers
296 views

Order of the Rubik's cube group

Associated to the Rubik's cube is a group as described in this Wikipedia article: $G = \langle F, B, U, L, D, R\rangle$. For example, the element $F$ corresponds to rotating the front face clockwise ...
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1answer
164 views

Where Does F' in Rubik's Cube Group Singmaster Notation Come From?

Basic 90° : F turns the front clockwise 180° : F^2 turns the front clockwise twice -90° : F' turns the front counter-clockwise Why are we calling the -90° rotation F' and not -F? (source: http://en....
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87 views

Rubik's cube computing algorithm

Is it possible to solve a 3x3 Rubik's cube using a tree similar to a 'mini-max game tree' (listing all possible unique moves, then listing further moves from this and so on), or is the sample space ...
2
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0answers
163 views

Minimum number of stickers required for 3x3 Rubik's Cube

Lately, the stickers on my V-Cube have been peeling off, and I became curious: what is the minimum number of stickers I would actually need in order to represent each unique configuration? I have but ...
6
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1answer
449 views

Miniature Rubik's Cube

$R$ is a clock-wise twist of a right-hand face and $U$ is a clock-wise twist of the upper face of a $2 \times 2 \times 2$ Rubik's Cube. Find the order of the subgroup generated by $\{R, U\}$
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1answer
72 views

How to use group theory to solve larrys square iphone app

There's a 2 d version of rubiks cube on apple app store. How can group theory give an algorithm to solve the iPhone app:larry's square.
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2answers
262 views

Is there a proof, that for any Rubik's cube, after solving the first 2 layers, there will always be an even amount of oriented edges?

I've always wondered if there was a proof for this, since the beginners method to solving the last layer depends on this (you either have no edges oriented correctly, an L, a line or a cross, and each ...
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1answer
299 views

Complex automata Rubiks cube question (with picture) help needed

Question 1 A Rubik’s Cube is a puzzle in the shape of a cube. Each face is covered by nine stickers, each of which is coloured with one of six colours: white, red, blue, orange, green, andyellow. An ...